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Introduction to Differential Equations

Consider the equation $f(x) = x^2 -3x + 1$. To solve this equation is to find the roots of $f$, which we can obtain with the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. For the example above, the roots are $x = \frac{3 + \sqrt{5}}{2}$ and $x = \frac{3 - \sqrt{5}}{2}$. Single variable functions such as the one previously mentioned merely contain one variable, $x$. However, in real-life applications, often times we are not dealing with such static functions. Instead, an equation may contain a rate of change. For example, consider an equation $P$ which measures the population at a given time $t$. Such an equation for $P$ might be:

(1)

\begin{align} P = \frac{dP}{dt} - 3t + 100 \end{align}

In such a case, we have an equation that contains a derivative. Such an equation is known as a differential equation.

Definition: A Differential Equation is an equation that contains some sort of derivative. An Ordinary Differential Equation (abbreviated as O.D.E.) is an equation that contains only regular derivatives, while a Partial Differential Equation (abbreviated as P.D.E.) is an equation that contains a partial derivative. The Order of a differential equation is the highest order regular or partial derivative present in the equation.

Differential equations appear in many applications ranging from biology to physics, and even economics as such equations often provide great models of our complex world.

Let's look at some examples of differential equations. One such first order differential equation is $y = y'$. One of the solutions to this differential equation is $y = e^x$ since $y' = e^x$. Another such solution is $y = 0$ since $y' = 0$. More generally, $y = Ce^x$ where $C \in \mathbb{R}$, is a solution to the differential equation $y = y'$.

Recall that a non-vertical line on the $xy$ plane can be given by the equation $y = mx + b$. Note that $y' = m$, which should intuitively make sense as the derivative of $y$ tells us the slope at any point on the curve generated by $y$, which in this case is a straight line with constant slope $m$. Therefore we can write the equation of a line in the form $y = xy' + b$. Note that any non-vertical line $y = mx + b$ is a solution to this differential equation.

Another example of a differential equation is Newton's second law in physics which states that $F = ma$ where $F$ is the net force of an object, $m$ is the mass of that object, and $a$ is the acceleration of that object. The net force $F$ is influenced by the force of gravity which is equal to the mass of the object $m$ multiplied by the acceleration from gravity near the surface of the earth, $g = \frac{9.8m}{s^2}$. There's also force as a result of air resistance which is equal to a specific drag constant multiplied by the velocity of that object, that is $\gamma v$, and so the force $F$ can be computed as:

(2)

\begin{align} F = mg - \gamma v \end{align}

Recall from calculus that at an arbitrary time $t$, that the derivative of the velocity of an objection is equal to the acceleration of that object at time $t$, that is $\frac{dv}{dt} = a$, and so we get that:

(3)

\begin{align} mg - \gamma v = m \frac{dv}{dt} \end{align}

Therefore the equation $\frac{dv}{dt} = g - \frac{\gamma v}{m}$ is a differential equation that models a falling object near the surface of the earth.

In general, finding the solutions to a differential equation can be difficult. For example, consider the following differential equation:

(4)

\begin{align} \quad 2y + yy' = y'' \end{align}

Finding the solutions to this differential equation is much more complicated and it is not obvious what functions to even test. Clearly $y = 0$ is a solution, however, is it the only solution? We will develop methods for evaluating various types of differential equations.